CN107330145A - It is a kind of while considering the jointed rock slope analysis of Ultimate Lower Bound Limit of rock mass translation and turning effect - Google Patents
It is a kind of while considering the jointed rock slope analysis of Ultimate Lower Bound Limit of rock mass translation and turning effect Download PDFInfo
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Abstract
The present invention relates to a kind of while the Analysis of Bearing Capacity Lower Bound Limit of consideration rock mass translation and turning effect, belongs to Rock Slope Stability Analysis technical field.The present invention uses rigid block movement unit by its discrete geometrical system for rigid sillar+structural plane, using the normal force of structural plane, shearing, moment of flexure as unknown quantity, build and meet the static(al) permissible stress that rigid block movement unit considers translation, the equilibrium equation of turning effect, structural plane shear yielding condition, structural plane tensile yield condition and structure face rotation yield condition and static(al) boundary condition simultaneously, using over-loading coefficient as object function, Lower Bound Limit linear math plan model is set up；Linear math plan model is solved using interior-point algohnhm, while solving margin of safety coefficient using iterative algorithm, the over-loading coefficient of jointed rock slope or the lower limit solution and corresponding yield region of margin of safety coefficient is obtained.Definite conception of the present invention, computational accuracy are high, can be applied to jointed rock slope translation or rotate the Analysis of Bearing Capacity of destruction.
Description
Technical field
The present invention is a kind of while under the jointed rock slope analysis of Ultimate of consideration rock mass translation and turning effect
Limit method, it is more particularly to a kind of while considering rock mass discontinuity failure by shear, tensile failure and the Analysis of Bearing Capacity method for rotating destruction.
Belong to Rock Slope Stability Analysis technical field.
Background technology
The unstability of jointed rock slope is a kind of common geological disaster.At home and abroad highway, railway, mine and water conservancy water
The catastrophic failure of a large amount of jointed rock slope unstable failures is encountered in the engineerings such as electricity, such as domestic Tianshengqiao Second-cascade Hydropower Station side
Slope Inclining destruction, Instability of Side Slope of Jinchuan Open-pit Mine destruction etc..The bearing capacity and method for estimating stability of jointed rock slope are
Become the important technological problems that Geotechnical Engineering circle must be solved.
Jointed rock slope substantial amounts of joint, crack because inside is existed so that its mechanical characteristic is extremely complex.Joint
Rock side slope is commonly considered as the complication system being made up of sillar and structural plane, the motion mode of sillar generally can be divided into translation and
Rotate, variation is also presented in the failure mode of structural plane.Sillar translational slip destruction problem is relatively easy, and sillar, which is rotated, topples over brokenly
Bad problem is then more complicated, the rotation Inclining destruction of general sillar can be divided into bulk topple over, Bending And Toppling and block Bending And Toppling
Etc. type.Since 1950s, the research of rock mass translation destruction and rotation destruction has just been taken seriously.In order to solve rock
The stability analysis problem that body translation, rotation are toppled over, the stability of slope computational methods based on Limit Equilibrium Principle are proposed out,
And it is widely used in engineering.This method typically does some to sillar deformation model it is assumed that rock side slope
Gliding mass is divided into stable region, dumping ground or sliding area, but its reasonability is still to be tested to some specific questions.Thereafter, many numerical value
Method is proposed to rock mass translation, rotates the analysis calculating of destruction unstability, such as Finite Element, discrete element method, block
Body unit method, DDA etc..Meanwhile, rock side slope translation, the destruction physical experiments rotated are also widely used for engineering reality
Trample.
For current achievement in research, while considering translation, the Analysis of Bearing Capacity of the jointed rock slope of turning effect
The problem of field also awaits further inquiring into the presence of some.Specific deficiency is shown：
(1) many analysis methods can not consider translation and the turning effect of rock mass simultaneously, such as balance method of rigid-body limit, have
First method of limit etc. can only typically consider the translation effect of rock mass, but engineering practice show the turning effect of rock mass be it is extremely important,
Indispensable, such as：It is its chief destructive pattern that destruction is rotated for bedding joint rock side slope.
(2) pattern of the structural plane destruction in jointed rock slope is also very various, it is considered that including three kinds：Sliding is cut
Cut through bad, tensile failure and rotate destruction, but existing analysis method is difficult while considering these three failure modes.
(3) in Limit Analysis field, particularly Lower Bound Limit analysis field, research rock mass considers translation, turning effect simultaneously
Achievement in research it is less.
The present invention is based on project of national nature science fund project (grant numbers：51564026) research work, by plasticity pole
Limit analysis lower limit law theory, rigid block movement mesh discretization technology and Mathematical Planning means combine, and propose that one kind is examined simultaneously
Consider translation, the Lower Bound Limit of the rock side slope analysis of Ultimate of turning effect.
The content of the invention
It is an object of the invention to provide a kind of while considering that the jointed rock slope limit of rock mass translation and turning effect is held
Power analysis Lower Bound Limit is carried, a kind of new method is provided for stability analysis of jointed rock slope.
The present invention general principle be：It is using rigid block movement unit that its is discrete using jointed rock slope as research object
For the geometrical system of rigid sillar+structural plane, using the normal force of structural plane, shearing, moment of flexure as unknown quantity, divided based on plastic limit
Lower limit law theory is analysed, builds and meets simultaneously：Rigid block movement unit considers translation, the equilibrium equation of turning effect, structural plane shearing
Yield condition, structural plane tensile yield condition and structure face rotate yield condition and the static(al) permissible stress of static(al) boundary condition
, then using the over-loading coefficient of load as object function, set up and consider that the jointed rock slope of translation and turning effect is held simultaneously
Carry the Lower Bound Limit linear math plan model of power analysis；Then linear math plan model is solved using interior-point algohnhm,
Margin of safety coefficient can be solved using iterative algorithm simultaneously, can finally obtain the over-loading coefficient or margin of safety of jointed rock slope
The lower limit solution and corresponding yield region of coefficient.
The technology path of the present invention is as shown in Figure 1.
The jointed rock slope analysis of Ultimate Lower Bound Limit of rock mass translation and turning effect is considered while of the invention
Technical scheme carry out according to the following steps successively：
First, the calculating parameter of jointed rock slope is drafted
According to the actual conditions of jointed rock slope, the calculating parameter that Lower Bound Limit analysis needs is drafted, is mainly included：Rock matter
The geometric parameter of side slope, joint parameters, the parameter (unit weight, cohesiveness, angle of friction, tensile strength etc.) of rock mass and jointed material,
Slope load parameter information.
2nd, using rigid block movement mesh discretization jointed rock slope
The present invention uses rigid block movement mesh discretization jointed rock slope, and rigid block movement unit stress is as shown in Fig. 2 adjacent
The structural plane stress of unit is as shown in Figure 3.Wherein (x, y) is global coordinate；(nk,sk) it is to be tied between adjacent block unit i, j
Structure face k local coordinate system, nkFor the exterior normal direction of structural plane, skFor the tangential of structural plane；Rigid block movement unit i centres of form Ci
Upper effect has equivalent load force vectorfxiFor external force in the x-direction, fyiFor external force in the y-direction, miFor
External force moment of flexure (regulation is counterclockwise forward direction)；Structural plane k centres of form P between adjacent block unit i, jkUpper effect has interior force vectorOpposite side effect has its reaction force vectorNkIt is along structural plane exterior normal nkSide
To normal force (draw malleation bear), VkIt is along the tangential s of structural planekThe shearing (producing block to rotate counterclockwise as just) in direction, Mk
For the internal force moment of flexure of structural plane (regulation is positive counterclockwise).
After rigid block movement mesh discretization jointed rock slope, side slope becomes the several of rigid block movement unit+structural plane
What system, the present invention is with the centre of form P in rigid block movement cellular construction facekThe interior force vector of upper effectTo be unknown
Amount builds the static(al) permissible stress of rock side slope.In order to simplify calculating, the present invention makes the following assumptions：(1) assume that sillar is firm
Body, the destruction of rock side slope is only occurred on the structural plane between adjacent block unit；(2) it is ideal to assume structural plane material
Rigid-plastic material, (3) assume that structural plane is likely to occur failure by shear, tensile failure or rotates destruction, and (4) sillar occurs simultaneously
Translation or rotation.
3rd, set up and consider rock mass translation, the jointed rock slope Analysis of Bearing Capacity Lower Bound Limit mathematics rule of turning effect simultaneously
Draw model
The purpose of the present invention is to solve for the lower limit solution of jointed rock slope over-loading coefficient or margin of safety coefficient, it is necessary to set up
Static(al) permissible stress.According to lower bound theorem, to set up the static(al) permissible stress of side slope need to meet equilibrium equation, yield condition
And boundary condition.
1st, object function
The overload of jointed rock slope is defined as by the inventive method：
K1=Fc/Fa (1)
In above formula, FcIt is ultimate load, FaIt is the external load of the currently practical application of edge slope structure, for rock side slope, asks pole
Its external load overloaded can be that deadweight can also be face power or concentrated force during limit state.
According to lower bound theorem, rock side slope is when reaching capacity state, the maximum of demand over-loading coefficient.The present invention will be super
It is as follows that load coefficient is set to object function：
Maximize:K1 (2)
2nd, rigid block movement unit equilibrium equation
If whole jointed rock slope by discrete for n rigid block movement unit,For the knot on any rigid block movement unit i
Structure face quantity.For any rigid block movement unit i in side slope shown in Fig. 2, its effect by various power and keep poised state,
The present invention considers translation, the turning effect of Rigid Body Element simultaneously, therefore considers the balance of power and the balance of torque simultaneously.For list
One its power of rigid block movement unit i and torque equilibrium equation are：
In above formula：N is the quantity of block in side slope；It is the structural plane quantity on rigid block movement unit block i；It is structural plane k centres of form PiThe interior force vector of upper effect；It is rigid block movement unit i shapes
Heart CiThe equivalent load force vector of effect；TkIt is structural plane k local coordinate (nk,sk) arrive global coordinate (x, y) conversion square
Battle array,αkIt is nkThe angle of axle and x-axis, counterclockwise for just；δ1It is structural plane k centres of form D in Fig. 2k
To point PkDistance；δ2It is D in sillar i in Fig. 2kPoint arrives centre of form CiThe distance arrived；K1For over-loading coefficient；K1For over-loading coefficient.
For the two-dimentional rock side slope containing n block, formula (3) includes the equilibrium equation and n equalising torque of 2n power
Equation, can write a Chinese character in simplified form as follows in the form of matrix, vector：
In above formula：C is to consider translation, the whole machine balancing matrix of turning effect,It is the interior force vector of all structural planes,
It is the equivalent load force vector of all Rigid Body Element centres of form.
3rd, the yield condition of structural plane
Present invention assumes that the destruction of jointed rock slope is occurred only on structural plane.At the same time consider sillar translation,
During rotational kinematics and dynamics effect, three kinds of failure modes (as shown in Figure 4) can occur for the structural plane between arbitrary neighborhood Rigid Body Element, including
Failure by shear, tensile failure and rotation are destroyed.
For any one structural plane k (as shown in Figure 3), in local coordinate system (nk,sk) in its centre of form effect have internal force
VectorThen the failure by shear Mohr-Coulomb yield conditions of structural plane are：
In above formula：nsIt is the quantity of structural plane in whole side slope, lkIt is structural plane k length, c is the cohesiveness of structural plane,It is the internal friction angle of structural plane, NkIt is structural plane k normal force (regulation is using pulling force as just), VkIt is structural plane k shearing.
The tensile yield condition of structural plane is：
Nk-σTlk≤ 0, k=(1 ..., ns) (6)
In above formula：nsFor the quantity of structural plane in whole side slope, lkFor structural plane k length, NkFor structural plane k normal direction
Power (regulation is using pulling force as just), σTFor the tensile strength of structural plane, because the joint of insertion can not typically bear tension, therefore
The tensile strength of structural plane takes σ in the present inventionT=0.
The shear yielding conditional (5) and tensile yield conditional (6) in integrated structure face, the then drawing that can obtain structural plane are cut
The amendment Mohr-Coulomb yield conditions of destruction are as shown in Figure 5.
When the structural plane of adjacent block unit rotates destruction, the stress that Rigid Body Element rotates under limiting condition
As shown in fig. 6, due to internal force moment MkDirection have two kinds of possible directions (clockwise and anticlockwise), therefore Rigid Body Element has
May rotating around structural plane two end points P1、P2Occur two kinds and rotate destruction.The rotation yield condition of structural plane as shown in fig. 7,
It can be written as：
In above formula：nsIt is the quantity of structural plane in whole side slope, lkIt is structural plane k length, NkIt is structural plane k normal direction
Power (regulation is using pulling force as just), MkIt is structural plane k internal force moment of flexure.
Composite type (5), formula (6), formula (7), obtain considering Rigid Body Element translation, the generalized yield bar of rotational kinematics and dynamics effect
Part：
In above formula：K=(1 ..., ns),For with force vector in structural planeThe structure of expression
Face k generalized yield condition.
Formula (9) amounts to 5nsIndividual constraints, 5nsIndividual constraint is Linear Constraints.By generalized yield conditional (8)
Write a Chinese character in simplified form as follows：
In above formula：F is generalized yield condition,For force vector in all structural planes.
4th, static(al) boundary condition
According to lower bound theorem, static(al) permissible stress field must is fulfilled for known static(al) boundary condition.Consider joint Yan Zhi sides
Known boundary condition is in slopeInterface b, its boundary condition expression formula is：
In above formula：It is the interior force vector of the structural plane of known boundaries condition,It is to make on known boundaries construction of condition face
Power and torque.
5th, while considering rock mass translation, the jointed rock slope Analysis of Bearing Capacity Lower Bound Limit mathematical programming model of turning effect
According to lower bound theorem, combined objective function formula (2), equilibrium equation (4), generalized yield conditional (10), border
Conditional (11), then can obtain and consider rock mass translation, the jointed rock slope Analysis of Bearing Capacity lower limit normal of turning effect simultaneously
Property mathematical programming model：
Above formula is a linear math plan model, and the over-loading coefficient of jointed rock slope can be arrived by being solved by optimized algorithm
With corresponding yield region.
4th, optimized algorithm solves the ultimate bearing capacity of jointed rock slope
Established by above-mentioned steps while considering rock mass translation, the jointed rock slope ultimate bearing capacity point of turning effect
The Lower Bound Limit mathematical programming model (formula (11)) of analysis, it is an extensive linear math plan model.The present invention is used
Fortran 12.0 writes Lower Bound Limit calculation procedure, and calculation procedure includes pre-treatment, builds Lower Bound Limit mathematical programming model, optimization
4 parts such as solution, post processing.The present invention is solved using interior-point algohnhm to linear math plan model.It can finally obtain
The lower limit solution and correspondence yield region of the over-loading coefficient of jointed rock slope stability.
For jointed rock slope, sometimes except over-loading coefficient also needs to solve margin of safety coefficient, it is of the invention general
Margin of safety coefficient is defined as：
In above formula：c,For the cohesiveness and angle of friction of structural plane, c',Coagulated for structural plane progress strength degradation is later
Poly- power and angle of friction, σT,σT'For the tensile strength that rock mass reduction is forward and backward.
If direct solution margin of safety COEFFICIENT K2, formula (9) can become nonlinear restriction, so as to cause formula (11) to be non-thread
Property plan model.In order to avoid solving Nonlinear programming Model, the present invention uses solution by iterative method margin of safety based on formula (11)
COEFFICIENT K2, specific method is：Strength of materials parameter is gradually reduced in formula (12) and formula (9) is brought into, is then followed using formula (11)
Ring solves K1, work as K2So that K1(error is less than 0.001) is the maximum of required margin of safety coefficient when being sufficiently close to 1
(specific iterative process refers to Fig. 1).
It is characteristic of the invention that：The inventive method for it is a kind of solve jointed rock slope margin of safety coefficient it is strict under
Limit method, this method effectively combines lower bound theorem, rigid block movement mesh discretization technology, linear math planning, with adjacent firm
Property Rigid Body Element between normal force, shearing and the moment of flexure of structural plane be unknown quantity, construct the static(al) permissible stress of Lower Bound Limit,
Static(al) permissible stress strictly meets three conditions：(1) equilibrium equation of power and torque, (2) structural plane failure by shear, stretching are broken
Bad and rotation destruction generalized yield condition, (3) static(al) boundary condition, and linear math plan model is entered using interior-point algohnhm
Row is solved, and has obtained jointed rock slope while considering sillar translation, turning effect, while considering structural plane failure by shear, drawing
Stretch destruction and rotate the limiting condition of three kinds of failure modes of destruction, obtain under corresponding over-loading coefficient or margin of safety coefficient
Limit solution and corresponding yield region.
The invention has the advantages that：
1st, the inventive method is a kind of strict Lower Bound Limit for solving jointed rock slope margin of safety coefficient, can be examined simultaneously
Consider translation, the turning effect of sillar, be a kind of analysis method for being closer to jointed rock slope practical distortion damage -form.
2nd, the present invention is using while consider the failure by shear of structural plane, tensile failure and rotation in jointed rock slope
Destruction, can obtain the yield region of three kinds of failure modes, more directly perceived to the damage -form of understanding side slope.
3rd, the inventive method build Lower Bound Limit model be linear programming model, can direct solution obtain side slope overload
The lower limit solution of coefficient, while the lower limit solution of solution by iterative method margin of safety coefficient can be used, mathematical model optimizing solves simple, accurate
Really.
4th, the inventive method definite conception, computational accuracy are high, are more suitable for relative to traditional balance method of rigid-body limit multiple
The stability analysis of miscellaneous jointed rock slope.
Brief description of the drawings
Fig. 1 is the technology of the present invention route map；
Fig. 2 is rigid block movement unit stress diagram；
Fig. 3 is structural plane stress diagram between adjacent rigid Rigid Body Element；
Fig. 4 is structural plane failure mode between adjacent rigid Rigid Body Element：Failure by shear, tensile failure and rotation destruction；
Fig. 5 is the amendment Mohr-Coulomb yield criterions that structural plane shear tension is destroyed；
Structural plane limiting condition stress diagram when Fig. 6 is Rigid Body Element Inclining destruction；
Fig. 7 is that structural plane rotates destruction yield condition schematic diagram；
Fig. 8 is the geometry schematic diagram of the jointed rock slope of embodiment 1；
Fig. 9 is the jointed rock slope rigid block movement mesh discretization schematic diagram of embodiment 1；
Figure 10 is yield region schematic diagram under the jointed rock slope limiting condition of embodiment 1；
Figure 11 is the geometry schematic diagram of the jointed rock slope of embodiment 2；
Figure 12 is the jointed rock slope rigid block movement mesh discretization schematic diagram of embodiment 2；
Figure 13 is yield region schematic diagram under the jointed rock slope limiting condition of embodiment 2.
Embodiment
The invention will be further described with specific embodiment below in conjunction with the accompanying drawings.
Embodiment 1：Embodiment 1 is outside the limit for solve a jointed rock slope containing 2 sillars using formula (11)
Load.
(1) calculating parameter of jointed rock slope is drafted.
Fig. 8 is the geometry schematic diagram of the jointed rock slope of embodiment 1.This side slope includes two sillars, the shape of sillar 1
Shape is rectangle, its wide 1.0m, high 2.0m；Sillar 2 is shaped as triangle, height 2.0m, bottom width 5.495m.Sillar 1 and sillar 2
Between be an insertion joint, joint plane inclination angle be 20 °.The physical and mechanical parameter of side slope material refers to table 1.In the centre of form of sillar 1
C1Place's effect has the external force F of a level to the left, and the purpose of the present embodiment is just to solve for F limiting value Fc。
The physical and mechanical parameter table of the jointed rock slope material of 1 embodiment of table 1
| Title material | Severe/(kNm-3) | Cohesion/kPa | Internal friction angle/(°) | Tensile strength/kPa |
| Intact rock | 25 | 4000.0 | 50.0 | 1000 |
| Joint plane | / | 1.0 | 30.0 | 0.0 |
(2) rigid block movement mesh discretization jointed rock slope is used
Using the jointed rock slope of rigid block movement mesh discretization embodiment 1, and with rigid block movement cellular construction face
Power (normal force, shearing, moment of flexure) is unknown quantity.The slope system covariance of embodiment 1 is 2 rigid block movement units (such as Fig. 9 institutes
Show).
(3) set up and consider rock mass translation, the Lower Bound Limit mathematics of the side slope Analysis of Bearing Capacity of embodiment 1 of turning effect simultaneously
Plan model
After jointed rock slope using the embodiment 1 of rigid block movement mesh discretization, solution is established according to formula (12)
Rock mass translation, the side slope bearing capacity Lower Bound Limit model of turning effect are considered simultaneously.
(4) optimized algorithm solves the ultimate bearing capacity of jointed rock slope
The Lower Bound Limit model set up according to formula (11), using the linear programming Optimization Solution program of establishment, is solved in side slope
The limiting value of external load F at the centre of form of block 1.Result of calculation is listed in Table 2 below.
Ultimate load of the embodiment 1 under two kinds of Failure Models is calculated in table 2, the first is only to consider block translation effect
The failure by shear answered, second is that block considers translation+turning effect simultaneously, and considers the failure by shear of structural plane simultaneously, draws
Stretch destruction, rotate destruction.It is 9673.70N, parsing that the inventive method, which calculates obtained ultimate load, under the first Failure Model
Solve as 9695.73N；It is 5742.61N, analytic solutions that the inventive method, which calculates obtained ultimate load, under second of Failure Model
For 5754.29N；The result that the inventive method is obtained respectively less than analytic solutions, belong to lower limit solution, and both errors are very small, checking
The correctness of the inventive method.In addition, the ultimate load of the first Failure Model is more than the limit lotus of second of Failure Model
Carry, illustrate for this rock side slope if only the failure by shear for considering translation, it will over-evaluate the bearing capacity of side slope, i.e.,：Turn
Dynamic destruction can occur prior to failure by shear.Figure 10 is that the inventive method under second of Failure Model of embodiment 1 calculates what is obtained
Shear yielding occurs for yield region schematic diagram, the joint plane in two Edge of block faces, while around P1Put the surrender that rotates.The present invention
Method considers translation, turning effect simultaneously, compared to the bearing capacity that conventional method can more accurately assess rock side slope.
The jointed rock slope ultimate load result of calculation of 2 embodiment of table 1
Embodiment 2：Embodiment 2 is that the intensity for solving a jointed rock slope containing 6 sillars using formula (12) is stored up
Standby coefficient.
(1) calculating parameter of jointed rock slope is drafted.
Figure 11 is the geometry schematic diagram of the jointed rock slope of embodiment 2.This 25 ° of side slope bottom surface inclination angle, side slope includes 6
Individual sillar, the physical dimension of each sillar refers to Figure 11.The physical and mechanical parameter of side slope material refers to table 3.In 6 sillars, rock
Block 1 is fixed sillar, immovable, and translation or rotation may occur for 2~sillar of sillar 6.The purpose of the present embodiment is to solve for rock
The margin of safety coefficient of matter side slope.
The physical and mechanical parameter table of the jointed rock slope material of 3 embodiment of table 2
| Title material | Severe/(kNm-3) | Cohesion/kPa | Internal friction angle/(°) | Tensile strength/kPa |
| Intact rock | 26 | 5000.0 | 55.0 | 1500 |
| Joint plane | / | 0.0 | 35.0 | 0.0 |
(2) rigid block movement mesh discretization jointed rock slope is used
Using the jointed rock slope of rigid block movement mesh discretization embodiment 2, and with rigid block movement cellular construction face
Power (normal force, shearing, moment of flexure) is unknown quantity.The slope system covariance of embodiment 2 is 6 rigid block movement units, such as Figure 12
It is shown.
(3) the Lower Bound Limit mathematics for the side slope Analysis of Bearing Capacity of embodiment 2 for considering rock mass translation and turning effect simultaneously is set up
Plan model
After jointed rock slope using the embodiment 2 of rigid block movement mesh discretization, solution is established according to formula (12)
Rock mass translation, the side slope bearing capacity Lower Bound Limit model of turning effect are considered simultaneously.
(4) optimized algorithm solves the ultimate bearing capacity of jointed rock slope
Due to current embodiment require that solve side slope margin of safety coefficient, therefore according to formula (12) set up Lower Bound Limit mould
Type, using linear programming Optimization Solution program solution by iterative method margin of safety coefficient.The margin of safety coefficient finally tried to achieve is
K2=1.435, corresponding Failure Model is that sillar 3, sillar 4, sillar 5, sillar 6 occur that anticlockwise to rotate surrender broken
It is bad, as shown in figure 13.Because block 1 is fixed, if therefore the present embodiment only considers the translation effect of sillar, this
Unstability will not occur for side slope.But the result calculated using the inventive method is：When margin of safety coefficient takes K2When=1.435, side
Inclining destruction counterclockwise occurs for slope.
Claims (3)
1. a kind of while considering the jointed rock slope analysis of Ultimate Lower Bound Limit of translation and turning effect, its feature exists
In：Using jointed rock slope as research object, rigid block movement unit is used by its discrete geometry system for rigid sillar+structural plane
System, using the normal force of structural plane, shearing, moment of flexure as unknown quantity, based on plastic limit analysis lower limit law theory, builds and meets simultaneously
Rigid block movement unit considers translation, the equilibrium equation of turning effect, structural plane shear yielding condition, structural plane tensile yield condition
Yield condition and the static(al) permissible stress of static(al) boundary condition are rotated with structural plane, then using the over-loading coefficient of load as mesh
Scalar functions, set up the Lower Bound Limit linear math planning for the jointed rock slope Analysis of Bearing Capacity for considering translation and turning effect simultaneously
Model；Then linear math plan model is solved using interior-point algohnhm, while solving margin of safety using iterative algorithm
Coefficient, the lower limit solution and corresponding yield region of the final over-loading coefficient or margin of safety coefficient for obtaining jointed rock slope.
2. it is according to claim 1 while under the jointed rock slope analysis of Ultimate of consideration translation and turning effect
Limit method, it is characterised in that：Comprise the following steps that：
(1) calculating parameter of jointed rock slope is drafted
According to the actual conditions of jointed rock slope, the calculating parameter that Lower Bound Limit analysis needs is drafted, including：Rock side slope it is several
What parameter, joint parameters, the parameter of slope load parameter information and rock mass and jointed material, the ginseng of the rock mass and jointed material
Number unit weight, cohesiveness, angle of friction, tensile strength；
(2) rigid block movement mesh discretization jointed rock slope is used
Using rigid block movement mesh discretization jointed rock slope, side slope becomes the geometrical system of rigid block movement unit+structural plane,
(x, y) is global coordinate；(nk,sk) between adjacent block unit i, j structural plane k local coordinate system, nkFor structural plane
Exterior normal direction, skFor the tangential of structural plane；Rigid block movement unit i centres of form CiUpper effect has equivalent load force vectorfxiFor external force in the x-direction, fyiFor external force in the y-direction, miFor external force moment of flexure, regulation is counterclockwise
It is positive；Structural plane k centres of form P between adjacent block unit i, jiUpper effect has interior force vectorOpposite side
Effect has its reaction force vectorNkIt is along structural plane exterior normal nkThe normal force in direction, draws malleation to bear,
VkIt is along the tangential s of structural planekThe shearing in direction, produces block and rotates counterclockwise as just, MkFor the internal force moment of flexure of structural plane, rule
Fixed is counterclockwise forward direction；
With the centre of form P in rigid block movement cellular construction facekThe interior force vector of upper effectRock matter is built for unknown quantity
The static(al) permissible stress of side slope；In order to simplify calculating, make the following assumptions：(1) it is rigid body, the destruction of rock side slope to assume sillar
Only occur on the structural plane between adjacent block unit (2) and assume that structural plane material is assumed for rigid-perfectly plastic material (3)
Structural plane is likely to occur failure by shear, tensile failure or rotates destruction (4) sillar while occurring translation or rotation；
(3) set up and consider rock mass translation and the jointed rock slope Analysis of Bearing Capacity Lower Bound Limit Mathematical Planning of turning effect simultaneously
Model
According to lower bound theorem, to set up the static(al) permissible stress of side slope need to meet equilibrium equation, yield condition and boundary condition；
(1) object function
Over-loading coefficient is set to object function as follows：Maximize:K1, the over-loading coefficient of jointed rock slope is defined as：K1=
Fc/Fa；
In above formula, FcIt is ultimate load, FaIt is the external load of the currently practical application of edge slope structure, for rock side slope, finding limit shape
The external load that it overloads during state is deadweight, face power or concentrated force；
(2) rigid block movement unit equilibrium equation
If whole jointed rock slope by discrete for n rigid block movement unit,For the structural plane on any rigid block movement unit i
Quantity, for any rigid block movement unit i in side slope, its effect by various power and keep poised state, while considering power
Balance and torque balance, be for its power of single rigid block movement unit i and torque equilibrium equation：
<mrow>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<msubsup>
<mi>m</mi>
<mi>i</mi>
<mi>k</mi>
</msubsup>
</munderover>
<msub>
<mover>
<mi>T</mi>
<mo>&RightArrow;</mo>
</mover>
<mi>k</mi>
</msub>
<mo>&CenterDot;</mo>
<msub>
<mover>
<mi>Q</mi>
<mo>&RightArrow;</mo>
</mover>
<mi>k</mi>
</msub>
<mo>+</mo>
<msub>
<mi>K</mi>
<mn>1</mn>
</msub>
<mover>
<msub>
<mi>F</mi>
<mi>i</mi>
</msub>
<mo>&RightArrow;</mo>
</mover>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
In above formula：N is the quantity of block in side slope；It is the structural plane quantity on rigid block movement unit block i；It is structural plane k centres of form PkThe interior force vector of upper effect；It is rigid block movement unit i shapes
Heart CiThe equivalent load force vector of effect；TkIt is structural plane k local coordinate (nk,sk) arrive global coordinate (x, y) conversion square
Battle array,αkIt is nkThe angle of axle and x-axis, counterclockwise for just；δ1It is structural plane k centres of form D in Fig. 2k
To point PkDistance；δ2It is D in sillar i in Fig. 2kPoint arrives centre of form CiThe distance arrived；K1For over-loading coefficient；
For the two-dimentional rock side slope containing n block, equilibrium equation writes a Chinese character in simplified form as follows in the form of matrix, vector：
<mrow>
<mi>C</mi>
<mover>
<mi>Q</mi>
<mo>&RightArrow;</mo>
</mover>
<mo>+</mo>
<msub>
<mi>K</mi>
<mn>1</mn>
</msub>
<mover>
<mi>F</mi>
<mo>&RightArrow;</mo>
</mover>
<mo>=</mo>
<mn>0</mn>
</mrow>
In above formula：C is the whole machine balancing matrix for considering translation and turning effect,It is the interior force vector of all structural planes,It is institute
There is the equivalent load force vector of the Rigid Body Element centre of form；
(3) yield condition of structural plane
Assuming that the destruction of jointed rock slope is occurred only on structural plane, translation and the rotational kinematics and dynamics effect of sillar are considered at the same time
At once, three kinds of failure modes, including failure by shear can occur for the structural plane between arbitrary neighborhood Rigid Body Element, tensile failure and turn
Dynamic destruction；
For any one structural plane k, in local coordinate system (nk,sk) in its centre of form effect have interior force vectorThen the failure by shear Mohr-Coulomb yield conditions of structural plane are：
In above formula：nsIt is the quantity of structural plane in whole side slope, lkIt is structural plane k length, c is the cohesiveness of structural plane,It is
The internal friction angle of structural plane, NkIt is structural plane k normal force, regulation is using pulling force as just, VkIt is structural plane k shearing；
The tensile yield condition of structural plane is：
Nk-σTlk≤ 0, k=(1 ..., ns)
In above formula：nsFor the quantity of structural plane in whole side slope, lkFor structural plane k length, NkFor structural plane k normal force, rule
It is fixed using pulling force as just, σTFor the tensile strength of structural plane, because the joint of insertion can not typically bear tension, therefore in structure
The tensile strength in face takes σT=0；
When the structural plane of adjacent block unit rotates destruction, Rigid Body Element rotates and is in limiting condition, due to
Internal force moment MkDirection have two kinds of directions clockwise and anticlockwise, therefore Rigid Body Element is rotating around two end points P of structural plane1、
P2Occur two kinds and rotate destruction, the rotation yield condition of structural plane is written as：
<mrow>
<mo>|</mo>
<msub>
<mi>M</mi>
<mi>k</mi>
</msub>
<mo>|</mo>
<mo>-</mo>
<msub>
<mi>N</mi>
<mi>k</mi>
</msub>
<mfrac>
<msub>
<mi>l</mi>
<mi>k</mi>
</msub>
<mn>2</mn>
</mfrac>
<mo>&le;</mo>
<mn>0</mn>
<mo>,</mo>
<mi>k</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>,</mo>
<mn>...</mn>
<mo>,</mo>
<msub>
<mi>n</mi>
<mi>s</mi>
</msub>
<mo>)</mo>
</mrow>
</mrow>
In above formula：nsIt is the quantity of structural plane in whole side slope, lkIt is structural plane k length, NkIt is structural plane k normal force, rule
It is fixed using pulling force as just, MkIt is structural plane k internal force moment of flexure；
It is more than composite type various, obtain considering the generalized yield condition of Rigid Body Element translation and rotational kinematics and dynamics effect：
In above formula：K=(1 ..., ns),It is with force vector in structural planeThe structural plane k's of expression
Generalized yield condition, writes a Chinese character in simplified form as follows：F is generalized yield condition,For force vector in all structural planes；
(4) static(al) boundary condition
According to lower bound theorem, static(al) permissible stress field must is fulfilled for known static(al) boundary condition, it is considered in jointed rock slope
Known boundary condition isInterface b, its boundary condition expression formula is： It is the structural plane of known boundaries condition
Interior force vector,It is the power and torque acted on known boundaries construction of condition face；
(5) while considering the jointed rock slope Analysis of Bearing Capacity Lower Bound Limit mathematical programming model of rock mass translation and turning effect
According to lower bound theorem, combined objective function, equilibrium equation, generalized yield condition and boundary condition are obtained while considering rock
Body translation and the jointed rock slope Analysis of Bearing Capacity Lower Bound Limit linear math plan model of turning effect：
<mfenced open = "" close = "}">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>M</mi>
<mi>a</mi>
<mi>x</mi>
<mi>i</mi>
<mi>m</mi>
<mi>i</mi>
<mi>z</mi>
<mi>e</mi>
<mo>:</mo>
<msub>
<mi>K</mi>
<mn>1</mn>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>S</mi>
<mi>u</mi>
<mi>b</mi>
<mi>j</mi>
<mi>e</mi>
<mi>c</mi>
<mi>t</mi>
<mi> </mi>
<mi>t</mi>
<mi>o</mi>
<mo>:</mo>
<mi>C</mi>
<mover>
<mi>Q</mi>
<mo>&RightArrow;</mo>
</mover>
<mo>+</mo>
<msub>
<mi>K</mi>
<mn>1</mn>
</msub>
<mover>
<mi>F</mi>
<mo>&RightArrow;</mo>
</mover>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mover>
<mi>Q</mi>
<mo>&RightArrow;</mo>
</mover>
<mo>)</mo>
</mrow>
<mo>&le;</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>Q</mi>
<mo>&RightArrow;</mo>
</mover>
<mi>b</mi>
</msub>
<mo>=</mo>
<mover>
<mi>Q</mi>
<mo>&OverBar;</mo>
</mover>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
Above formula is linear math plan model, the over-loading coefficient that obtains jointed rock slope is solved by optimized algorithm and right
The yield region answered；
(4) optimized algorithm solves the ultimate bearing capacity of jointed rock slope
Linear math plan model is solved using interior-point algohnhm, the over-loading coefficient of jointed rock slope stability is obtained
Lower limit solution and correspondence yield region.
3. it is according to claim 1 while consider the rock side slope Analysis of Bearing Capacity Lower Bound Limit of translation and turning effect, its
It is characterised by：The margin of safety coefficient of jointed rock slope is：
In above formula：c,For the cohesiveness and angle of friction of structural plane, c',The later cohesiveness of strength degradation is carried out for structural plane
And angle of friction, σT,σT'For the tensile strength that rock mass reduction is forward and backward；Using solution by iterative method margin of safety COEFFICIENT K2。
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| CN114065590B (en) * | 2021-11-25 | 2023-04-07 | 中国电建集团成都勘测设计研究院有限公司 | Underground cavern block stability analysis method fusing rigid body limit balance method and finite element method |
| CN114065590A (en) * | 2021-11-25 | 2022-02-18 | 中国电建集团成都勘测设计研究院有限公司 | Underground cavern block stability analysis method fusing rigid body limit balance method and finite element method |
| CN114491737A (en) * | 2021-12-30 | 2022-05-13 | 北京市政路桥股份有限公司 | Stability mechanical analysis method for cutting masonry retaining wall |
| CN114491737B (en) * | 2021-12-30 | 2023-07-14 | 北京市政路桥股份有限公司 | Stability mechanical analysis method of cutting masonry retaining wall |
| CN114417482A (en) * | 2022-02-16 | 2022-04-29 | 中信建筑设计研究总院有限公司 | Method for establishing simplified mechanical model of reinforced masonry infilled wall |
| CN114417482B (en) * | 2022-02-16 | 2024-05-07 | 中信建筑设计研究总院有限公司 | Method for establishing simplified mechanical model of reinforced masonry infilled wall |
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